Understanding Dynamic Programming: Fibonacci Sequence, Shortest Paths, and More

1. Chapter 8 Dynamic Programming

2. Fibonacci sequence • Fibonacci sequence: 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , … Fi = i if i 1 Fi = Fi-1 + Fi-2 if i 2 • Solved by a recursive program: • Much replicated computation is done. • It should be solved by a simple loop.

3. Dynamic Programming • Dynamic Programming is an algorithm design method that can be used when the solution to a problem may be viewed as the result of a sequence of decisions

4. The shortest path • To find a shortest path in a multi-stage graph • Apply the greedy method : the shortest path from S to T : 1 + 2 + 5 = 8

5. The shortest path in multistage graphs • e.g. • The greedy method can not be applied to this case: (S, A, D, T) 1+4+18 = 23. • The real shortest path is: (S, C, F, T) 5+2+2 = 9.

6. Dynamic programming approach • Dynamic programming approach (forward approach): • d(S, T) = min{1+d(A, T), 2+d(B, T), 5+d(C, T)} • d(A,T) = min{4+d(D,T), 11+d(E,T)} = min{4+18, 11+13} = 22.

7. Dynamic programming • d(B, T) = min{9+d(D, T), 5+d(E, T), 16+d(F, T)} = min{9+18, 5+13, 16+2} = 18. • d(C, T) = min{ 2+d(F, T) } = 2+2 = 4 • d(S, T) = min{1+d(A, T), 2+d(B, T), 5+d(C, T)} = min{1+22, 2+18, 5+4} = 9. • The above way of reasoning is called backward reasoning.

8. Backward approach (forward reasoning) • d(S, A) = 1 d(S, B) = 2 d(S, C) = 5 • d(S,D)=min{d(S, A)+d(A, D),d(S, B)+d(B, D)} = min{ 1+4, 2+9 } = 5 d(S,E)=min{d(S, A)+d(A, E),d(S, B)+d(B, E)} = min{ 1+11, 2+5 } = 7 d(S,F)=min{d(S, A)+d(A, F),d(S, B)+d(B, F)} = min{ 2+16, 5+2 } = 7

9. d(S,T) = min{d(S, D)+d(D, T),d(S,E)+ d(E,T), d(S, F)+d(F, T)} = min{ 5+18, 7+13, 7+2 } = 9

10. Principle of optimality • Principle of optimality: Suppose that in solving a problem, we have to make a sequence of decisions D1, D2, …, Dn. If this sequence is optimal, then the last k decisions, 1  k  n must be optimal. • e.g. the shortest path problem If i, i1, i2, …, j is a shortest path from i to j, then i1, i2, …, j must be a shortest path from i1 to j • In summary, if a problem can be described by a multistage graph, then it can be solved by dynamic programming.

11. Dynamic programming • Forward approach and backward approach: • Note that if the recurrence relations are formulated using the forward approach then the relations are solved backwards . i.e., beginning with the last decision • On the other hand if the relations are formulated using the backward approach, they are solved forwards. • To solve a problem by using dynamic programming: • Find out the recurrence relations. • Represent the problem by a multistage graph.

12. The resource allocation problem • m resources, n projects profit p(i, j) : j resources are allocated to project i. maximize the total profit.

13. The multistage graph solution • The resource allocation problem can be described as a multistage graph. • (i, j) : i resources allocated to projects 1, 2, …, j e.g. node H=(3, 2) : 3 resources allocated to projects 1, 2.

14. Find the longest path from S to T : (S, C, H, L, T), 8+5+0+0=13 2 resources allocated to project 1. 1 resource allocated to project 2. 0 resource allocated to projects 3, 4.

15. The traveling salesperson (TSP) problem • e.g. a directed graph : • Cost matrix:

16. The multistage graph solution • A multistage graph can describe all possible tours of a directed graph. • Find the shortest path: (1, 4, 3, 2, 1) 5+7+3+2=17

17. The representation of a node • Suppose that we have 6 vertices in the graph. • We can combine {1, 2, 3, 4} and {1, 3, 2, 4} into one node. • (3),(4,5,6) means that the last vertex visited is 3 and the remaining vertices to be visited are (4, 5, 6).

18. The dynamic programming approach • Let g(i, S) be the length of a shortest path starting at vertex i, going through all vertices in S and terminating at vertex 1. • The length of an optimal tour : • The general form: • Time complexity: ( ),( ) (n-k)  (n-1)

19. The longest common subsequence (LCS) problem • A string : A = b a c a d • A subsequence of A: deleting 0 or more symbols from A (not necessarily consecutive). e.g. ad, ac, bac, acad, bacad, bcd. • Common subsequences of A = b a c a d and B = a c c b a d c b : ad, ac, bac, acad. • The longest common subsequence (LCS) of A and B: a c a d.

20. The LCS algorithm • Let A = a1 a2 am and B = b1 b2 bn • Let Li,j denote the length of the longest common subsequence of a1 a2 ai and b1 b2 bj. • Li,j = Li-1,j-1 + 1 if ai=bj max{ Li-1,j, Li,j-1 } if aibj L0,0 = L0,j = Li,0 = 0 for 1im, 1jn.

21. The dynamic programming approach for solving the LCS problem: • Time complexity: O(mn)

22. Tracing back in the LCS algorithm • e.g. A = b a c a d, B = a c c b a d c b • After all Li,j’s have been found, we can trace back to find the longest common subsequence of A and B.

23. 0/1 knapsack problem • n objects , weight W1, W2, ,Wn profit P1, P2, ,Pn capacity M maximize subject to M xi = 0 or 1, 1in • e. g.

24. The multistage graph solution • The 0/1 knapsack problem can be described by a multistage graph.

25. The dynamic programming approach • The longest path represents the optimal solution: x1=0, x2=1, x3=1 = 20+30 = 50 • Let fi(Q) be the value of an optimal solution to objects 1,2,3,…,i with capacity Q. • fi(Q) = max{ fi-1(Q), fi-1(Q-Wi)+Pi } • The optimal solution is fn(M).

26. Optimal binary search trees • e.g. binary search trees for 3, 7, 9, 12;

27. Optimal binary search trees • n identifiers : a1 <a2 <a3 <…< an Pi, 1in : the probability that ai is searched. Qi, 0in : the probability that x is searched where ai < x < ai+1 (a0=-, an+1=).

28. Identifiers : 4, 5, 8, 10, 11, 12, 14 • Internal node : successful search, Pi • External node : unsuccessful search, Qi • The expected cost of a binary tree: • The level of the root : 1

29. The dynamic programming approach • Let C(i, j) denote the cost of an optimal binary search tree containing ai,…,aj . • The cost of the optimal binary search tree with ak as its root :

30. General formula

31. Computation relationships of subtrees • e.g. n=4 • Time complexity : O(n3) when j-i=m, there are (n-m) C(i, j)’s to compute. Each C(i, j) with j-i=m can be computed in O(m) time.

32. Matrix-chain multiplication • n matrices A1, A2, …, An with size p0  p1, p1  p2, p2  p3, …, pn-1  pn To determine the multiplication order such that # of scalar multiplications is minimized. • To compute Ai  Ai+1, we need pi-1pipi+1 scalar multiplications. e.g. n=4, A1: 3  5, A2: 5  4, A3: 4  2, A4: 2  5 ((A1 A2)  A3)  A4, # of scalar multiplications: 3 * 5 * 4 + 3 * 4 * 2 + 3 * 2 * 5 = 114 (A1 (A2 A3))  A4, # of scalar multiplications: 3 * 5 * 2 + 5 * 4 * 2 + 3 * 2 * 5 = 100 (A1 A2)  (A3 A4), # of scalar multiplications: 3 * 5 * 4 + 3 * 4 * 5 + 4 * 2 * 5 = 160

33. Let m(i, j) denote the minimum cost for computing Ai  Ai+1  … Aj • Computation sequence : • Time complexity : O(n3)